Universal origin of heavy elements

The abundance of heavy elements through the rapid neutron capture process or r -process is intimately related to the competition between neutron capture and β decay rates, which ul-timately depends on the binding energy of atomic nuclei. The well-known Bethe-Weizs¨acker semi-empirical mass formula 1,2 describes the binding energy of ground states in nuclei with temperatures of T ≈ 0 MeV, where the nuclear symmetry energy saturates between 23 − 26 MeV. Here we ﬁnd a larger saturation energy of ≈ 30 MeV for nuclei at T ≈ 0 . 7 − 1 . 3 MeV,

which corresponds to the typical temperatures where seed elements are created during the cooling down of the ejecta following neutron-star mergers 3 and collapsars 4 . This large symmetry energy yields a reduction of the binding energy per nucleon for neutron-rich nuclei; hence, the close in of the neutron dripline, where nuclei become unbound. This finding constrains exotic paths in the nucleosynthesis of heavy elements -as supported by microscopic calculations of radiative neutron-capture rates 5  Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to jnorce@uwc.ac.za.
The binding energy of a nucleus with Z protons and N neutrons can be described by the well-known Bethe-Weizsäcker semi-empirical mass formula (SEMF) 1, 2 , where A = Z +N is the mass number and a v , a s , a c , a sym and a p are the volume, surface, Coulomb, symmetry energy and pairing coefficients, respectively. The symmetry energy, a sym (A)(N − Z) 2 /A, reduces the total binding energy B(Z, A) of a nucleus as the neutron-proton asymmetry becomes larger, i.e. for N ≫ Z, and yields the typical negative slope of the binding energy curve 6 for A > 62. It is divided by A to reduce its importance for heavy nuclei, and it depends on the mass dependency of a sym (A). Its convergence for heavy nuclei establishes the frontiers of the neutron dripline for particle-unbound nuclei and eventually leads to the disappearance of protons at extreme nuclear densities 7 .
Additionally, a sym (A) is relevant to understanding neutron skins 8 , the effect of three-nucleon forces 9 and -through the equation of state (EoS) -supernovae cores, neutron stars and binary mergers [10][11][12] . The latter are the first known astrophysical site where heavy elements are created through the rapid neutron-capture or r-process 5,13 . The identification of heavy elements in neutron star mergers is supported by the short duration gamma-ray bursts via their infrared afterglow 14only understood by the opacities of heavy nuclei -as well as blueshifted Sr II absorption lines 15 , following the expansion speed of the ejecta gas at v = 0.1 − 0.3 c. Mergers are expected to be the only source for the creation of elements above lead and bismuth, as inferred from the very scarce abundance of actinides in the solar system 16 . Other potential sources of heavy elements involve different types of supernova (e.g. collapsars 4 -the supernova-triggering collapse of rapidly rotating massive stars -and type-II supernova 5 ), which need to be considered to elucidate the universality of r-process abundances 17 in extremely metal-poor stars 18 .
It is the motivation of this work to understanding the limits of the neutron dripline and heavyelement production through the r-process by investigating a sym (A) at different temperatures using available data; namely, photo-absorption cross sections, binding energies and giant dipole resonances. Generally, a sym (A) is parametrized using the leptodermous approximation of Myers and which considers the modification of the volume symmetry energy, S v , by the surface symmetry energy S s .
The giant dipole resonance (GDR) represents the main contribution to the absorption and emission of electromagnetic radiation (photons) in nuclei 20 . The dynamics of this quantum collective excitation is characterized by the inter-penetrating motion of proton and neutron fluids out of phase 21 , which results from the density-dependent symmetry energy, a sym (A)(ρ N − ρ Z ) 2 /ρ, acting as a restoring force 20 ; where ρ N , ρ Z and ρ = ρ N + ρ Z are the neutron, proton and total density, respectively, which spread uniformly throughout the nucleus.
The ratio of the induced dipole moment to an applied constant electric field yields the static nuclear polarizability, α. Using the hydrodynamic model and assuming inter-penetrating proton and neutron fluids with a well-defined nuclear surface of radius R = r 0 A 1/3 fm and ρ Z as the potential energy of the liquid drop, Migdal 21 obtains the following relation between the static nuclear polarizability, α, and a sym , where r 0 = 1.2 fm, e 2 = 1.44 MeV fm in the c.g.s. system, and a constant value of a sym = 23 MeV was utilized.
Alternatively, α can be calculated for the ground states of nuclei using second-order perturbation theory 22 following the sum rule, where E γ is the γ-ray energy corresponding to a transition connecting the ground state |i and an excited state |n , M the nucleon mass, f in the dimensionless oscillator strength for E1 transitions 22 and σ −2 the second moment of the total electric-dipole photo-absorption cross section, where σ total (E γ ) is the total photo-absorption cross section, which generally includes (γ, n) + (γ, pn) + (γ, 2n) + (γ, 3n) photoneutron and scarcely available photoproton cross sections 23 , in competition in the GDR region 24,26 . By comparing Eqs. 3 and 6, a mass-dependent symmetry energy, a sym (A), is extracted in units of MeV, Empirical evaluations reveal that σ −2 can also be approximated by σ −2 = 2.4κA 5/3 , where the dipole polarizability parameter κ measures GDR deviations between experimental and hydrodynamic model predictions 27 .
where K is the real eigenvalue of ▽ 2 ρ Z +K 2 ρ Z = 0, with the boundary condition (n▽ρ Z ) surf ace = 0, and has a value of KR = 2.082 for a spherical nucleus 35 . For quadrupole deformed nuclei with an eccentricity of a 2 − b 2 = ǫR 2 , where a and b are the half axes and ǫ the deformation parameter, the GDR splits into two peaks with similar values of Ka and Kb ≈ 2.08 34 . Henceforth, we propose the symmetrized relation for deformed nuclei, The GDR cross-section data for each nucleus was obtained from the EXFOR and ENDF databases 36, 37 and fitted with one or two Lorentzian curves to extract E GDR and Γ GDR , as shown e.g. in Fig. 2 for 208 Pb. The data set for each nucleus was selected based on the number of data points, experimental method and energy range. In this work, the maximum integrated γ-ray energy,  Furthermore, it is interesting to investigate the behavior of a sym (A) using the available information on GDRs built on excited states, below the critical temperatures and spins where the GDR width starts broadening; i.e. for moderate average temperatures of T T c = 0.7 + 37.5/A MeV and spins J below the critical angular momentum J J c = 0.6A 5/6 . In fact, similar centroid energies, E exc GDR , and resonance strengths, S exc GDR -relative to the Thomas-Reiche-Kuhn E1 sum rule 22 -to those found for the ground-state counterparts 24, 25 indicate a common physical origin for all GDRs, in concordance with the Brink-Axel hypothesis that assumes that a GDR can be built on every state in a nucleus 39,40 .
Applying again Eqs. 9 and 10, the right panel of Fig. 3    showing average binding energies per nucleon using the Bethe-Weizsäcker SEMF for a sym = 23 MeV (left) and a sym = 30 MeV (right). Atomic masses in the bottom panels are extracted from the 2020 atomic mass evaluation (AME 2020) 46 .
Finally, the effects from the larger symmetry energy determined in this work are shown in